Single peak solutions for an elliptic system of FitzHugh–Nagumo type

Abstract

We study the Dirichlet problem for an elliptic system derived from FitzHugh–Nagummo model as follows:

$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u =f(u)- v, \qquad&\text {in}\ \Omega ,\\&-\Delta v+\gamma v =\delta _\varepsilon u,&\text{ in }\ \Omega ,\\&u=v =0,&\text {on}\ \partial \Omega , \end{aligned} \right. \end{aligned}$$

where $\Omega$ represents a bounded smooth domain in $\mathbb {R}^2$ and $\varepsilon , \gamma$ are positive constants. The parameter $\delta _{\varepsilon }>0$ is a constant dependent on $\varepsilon$, and the nonlinear term f(u) is defined as $u(u-a)(1-u)$. Here, a is a function in $C^2(\Omega )\cap C^1({\overline{\Omega }})$ with its range confined to $(0,\frac{1}{2})$. Our research focuses on this spatially inhomogeneous scenario whereas the scenario that a is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov–Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on a, which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment.